It is almost unnecessary to observe, that the instructions
we have given for the drawing of the anterior face F' G' of
the wheel, are equally applicable to the posterior FT T
which is parallel to it, and in all respects the same; the
common centre of all the circles in it being at O', Fig. 4,
is projected to 0 in Fig. 3. Hence, it will be easy to
construct the ellipses representing these circles in the
oblique projection, and consequently to determine the
points e, f Jc, &c., in the curvature of the teeth; observing
that, as their centre lines converge to C in the front face,
they all tend to 0 in the remoter surface, which is, how-
ever, for the most part concealed by the former.

It would be superfluous to enter into any details re-
garding the construction of the oblique view of the rim,
eye and arms, which are drawn upon precisely similar
principles to those we have already so fully explained.

Projections of Eccentrics.—Plate XYIII.

The term eccentric is applied in general to all such
curves as are composed of points situated at unequal dis-
tances from a central point or axis. The ellipse, the curve
called the heart, and even the circle itself when supposed
to be fixed upon an axis which does not pass through its
centre, are examples of eccentric curves.

The object of such curves, which are of frequent occur-
rence in machinery, is to convert a rotatory into an alter-
nating rectilinear motion ; and their forms admit of an
infinite variety, according to the nature of the motion de-
sired to be imparted. Examples of their application occur
in many arrangements of pumps, presses, valves of steam-
engines, spinning and weaving machines, &c.

Fig. 1. To draw the eccentric symmetrical curve called
the heart, which is such as, when revolving with a uni-
form motion on its axis, to communicate to a movable
point A, a uniform rectilinear motion of ascent and
descent

Let C be the axis or centre of rotation upon which
the eccentric is fixed, and which is supposed to revolve
uniformly; and let A A' be the distance which the
point A is required to traverse during a half revolution
of the eccentric. From the centre C, with radii re-
spectively equal to C A and C A', describe two circles ;
divide the greatest into any number of equal parts,
(say 16), and draw through these points of division the
radii C 1, C 2, C 3, &c. Then, divide the line A A' into
the same number of equal parts as are contained in the
semicircle, (that is, into 8, in the example now before us),
and through all the points V, 21, S', &c., draw circles con-
centric with the former ; the points of their intersection
B, D, E, &c., with the respective radii C 1, C 2, C 3, &c.
are points in the curve required, its vertex being at the
point 8.

It will now be obvious that when the axis, in its ano-u-
lar motion, shall have passed through one division, in
other words, when the radius C 1 coincides with C A', the
point A, being urged upwards by the curvature of the
revolving body on which it rests, will have taken the
position indicated by 1' ; and further, when the succeed-

ing radius C 2 shall have assumed the same position, the
point A will have been raised to 2' ; and so on till it
arrives at A', after a half revolution of the eccentric. The
remaining half A G F 8 of the eccentric, being exactly
symmetrical with the other, will enable the point A to
descend in precisely the same manner as it is elevated.
It is thus manifest that this curve is fitted to impress a
uniform motion upon the point A itself; but in practice a
small friction roller is usually interposed between the
surface of the eccentric and the piece which is to be actu-
ated by it. Accordingly the point A is to be taken as the
centre of this roller, and the curve whose construction we
have just explained is replaced by another similar to, and
equidistant from it, which is drawn tangentially to arcs
of circles described from the various points in the primary
curve with the radius of the roller. This second curve is
manifestly endowed with the same properties as the other;
for supposing the point e, for example, to coincide with A,
if we cause the axis to revolve through a distance equal to
one of the divisions, the point f which is the intersection
of the curve with the circle whose radius is C 1', will then
obviously have assumed the position 1'; at the next por-
tion of the revolution, the point g, (which is such that the
angle / C g is equal to e C /), will have arrived at 2', and
so on. Thus it is plain that the point a will be elevated
and depressed uniformly by means of the second curve, in
the same manner as that denoted by A is actuated by
the first.

It is worthy of remark that all the diameters A 8, B F,
D G, &c., of the eccentric are equal; this circumstance
may, in some instances, be taken advantage of by placing
two friction rollers diametrically opposite to each other,
which will thus be alternately and similarly impelled, and
so perform the functions of a spring.

Fig. 2 is an elevation, and Fig. 3 a vertical section of
an eccentric such as we have just described, in the form in
which it would be applied in practice.

Fin. 4. To draw a double eccentric curve which shall

o

impart a uniform motion of ascent and descent to the
point A, traversing an arc of a circle A A'.

First, divide the given arc A A' into any number of
equal parts, (8 in the present example), and from the com-
mon centre, or axis C of the eccentric, describe circles
passing through each of the points of division V, 2', S', &c.
Divide also the circle passing through 0, the centre of the
arc A A', into twice the number of equal parts; then taking
up in the compasses the length A 0, and placing one of the
points at the division marked 1, describe an arc of a circle,
which will cut at B the circle drawn with the radius Cl';
from the next point of division 2, mark off, in the same
manner, the point D in the circle whose radius is C 2',
and so on. The points B; D, E, &c., thus obtained, are
points in the curve required, which, supposing the eccen-
tric to revolve uniformly, will possess the property of com-
municating to the point A a uniform motion of ascent and
descent along the arc A A'. This admits of easy demon-
stration. The angle B C F is half of 21 C D, and conse-
quently, when the point B has arrived at 1', the radius